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Theorem axnul 4045
 Description: The Null Set Axiom of ZF set theory: there exists a set with no elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks, this is presented as a separate axiom; here we show it can be derived from Separation ax-sep 4038. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tells us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 4043). This proof, suggested by Jeff Hoffman (3-Feb-2008), uses only ax-5 1533 and ax-gen 1536 from predicate calculus, which are valid in "free logic" i.e. logic holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] p. 277). Thus our ax-sep 4038 implies the existence of at least one set. Note that Kunen's version of ax-sep 4038 (Axiom 3 of [Kunen] p. 11) does not imply the existence of a set because his is universally closed i.e. prefixed with universal quantifiers to eliminate all free variables. His existence is provided by a separate axiom stating (Axiom 0 of [Kunen] p. 10). See axnulALT 4044 for a proof directly from ax-rep 4028. This theorem should not be referenced by any proof. Instead, use ax-nul 4046 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by NM, 7-Aug-2003.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.)
Assertion
Ref Expression
axnul
Distinct variable group:   ,

Proof of Theorem axnul
StepHypRef Expression
1 ax-sep 4038 . 2
2 pm3.24 857 . . . . . 6
32intnan 885 . . . . 5
4 id 21 . . . . 5
53, 4mtbiri 296 . . . 4
65alimi 1546 . . 3
76eximi 1574 . 2
81, 7ax-mp 10 1
 Colors of variables: wff set class Syntax hints:   wn 5   wb 178   wa 360  wal 1532  wex 1537   wcel 1621 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-sep 4038 This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538
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